Note: I was assuming it is intensional that facing forwards you have a head turner, and when walking left or right you don't want/need a head turner. If that is the case, what you have is fine. You just need to finishing rigging up walking for the Left Profile (similar to the Right Profile).
Silly me. I just re-read your original question! Add a "Eye Gaze" behavior to your "Front profile", "Left Profile" and "Right Profile" under "Head". The eye gaze at the root of the puppet only controls one set of eyes. You need to add extra behaviors for the other profiles. (At least that worked for me when I added them.) That is why only the first profile worked - the root eye gaze behavior latched onto those eyes.
But I think you will want to increase the eyeball size - the eyes cannot move much at present because the eyeballs are so small compared to the pupil size. Sometimes an eye goes backwards as well, which I think happens when the pupil is outside of the eyeball size. See also Debugging Adobe Character Animator Eyes – Extra Ordinary, the Series for my general advice on eyes (but I have not updated to mention the "crazy eyes" effect yet).